Theorem crnggrpd 20212

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20211	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20207	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18921  CRingccrg 20199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-ring 20200  df-cring 20201

This theorem is referenced by:  fermltlchr  21495  psdmul  22109  psd1  22110  psdpw  22113  ply1chr  22249  ply1fermltlchr  22255  elrgspnsubrunlem1  33247  elrgspnsubrunlem2  33248  erlbr2d  33264  rlocaddval  33268  rloccring  33270  rloc0g  33271  rlocf1  33273  fracerl  33305  ressply1evls1  33583  evl1deg1  33594  evl1deg2  33595  evl1deg3  33596  ply1dg1rt  33597  vr1nz  33608  irngss  33733  irredminply  33755  algextdeglem4  33759  algextdeglem5  33760  aks6d1c1p3  42128  aks6d1c2lem4  42145  aks6d1c6lem2  42189  aks5lem2  42205  evlsbagval  42556  selvvvval  42575  evlselv  42577  selvadd  42578  prjcrv0  42623